Optimization of the Matrix Fourier-Filter for a Class of Nonlinear Optical Models with an Integral Objective Functional
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OPTIMIZATION OF THE MATRIX FOURIER-FILTER FOR A CLASS OF NONLINEAR OPTICAL MODELS WITH AN INTEGRAL OBJECTIVE FUNCTIONAL
S. V. Sazonova1 and A. V. Razgulin2
UDC 517.956.4, 517.977.58
We consider a new formulation of the Fourier-filtering problem that uses matrix Fourier-filters as the controls in nonlinear optical models described by quasi-linear functional-differential diffusion equations. Solvability of the control problem is proved for various classes of matrix Fourier-filters with a timeintegral objective functional. Differentiability of the functional with respect to the matrix Fourier-filter and convergence of a variant of the gradient projection method are proved. Examples of numerical simulation of controlled structure formation are presented, and the advantages of matrix Fourier-filters compared with traditional multiplier filters are demonstrated. Keywords: Fourier-filtering, functional-differential diffusion equation, Fourier-filter control, matrix Fourier-filter, functional, gradient, nonlinear optical models, feedback, numerical solution of optimization problem.
Introduction Fourier-filtering is a common signal processing method in various settings. Fourier-filtering involves changing the signal by manipulation of its Fourier-transforms. In optics, Fourier-filtering is performed, for instance, by a confocal 4 − f system of two thin lenses [1] with a spatial filter (a phase Zernike filter [2, 3], a “phase knife” filter [4], a “dark field” amplitude filter [5], and others) in the common focal plane. Such filters act on each Fourier-harmonic separately and are classified as multiplier filters. Fourier filters are the basis for phase visualization methods and optical computation [6, 7], adaptive distortion suppression and high-resolution wavefront correction [2, 3], formation and stabilization of optical structures with specified properties [5, 8, 9 and others]. The development of light-modulator technology with control microchips [3, 10] enables us to restrict the analysis to finite-aperture optics and discrete Fourier filters. The corresponding models of controlled discrete Fourierfiltering in nonlinear optical feedback systems have been developed in [11, 14]; bounds on the rate of convergence of projection-difference approximations for control problems with a multiplier filter have been obtained in [12]. Application of a wider class of filters constitutes a promising direction for the development of Fourier-filtering methods. A new formulation of the Fourier-filtering problem [13] uses matrix Fourier filters instead of the traditional multiplier filters, examines in detail the properties of the matrix Fourier-filtering operator, and studies the matrix-filter control problem with a terminal objective function. The aim of this study is to investigate the matrix-filter control problem with an integral weight functional and to compare the optimization efficiency on classes of matrix filters and multiplier filters. 1
Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University,
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